Can I upload a fillable PDF form to my website and allow people to email it to me directly from the form?

Yes, you can add a button with a mouse up javascript event hands off the PDF to the default email program.

Note: Adobe Reader is required to submit the PDF form

Check out the following web tool to help you generate the button action script:

fdftoolkit.net/email-pdf-mailto-generator.aspx

Here's a javascript example:

//Place this script in the buttons mouseUp JavaScript event 
//http://www.adobe.com/content/dam/Adobe/en/devnet/acrobat/pdfs/js_api_reference.pdf#page=345 
//SUBMIT PDF FORMAT WITH ADOBE READER 
var url = "mailto:you@domain.com?subject=SUBJECT&body=BODY&cc=&bcc="; 
var submitAs = "PDF";//PDF,FDF,XFDF,XDP,XML 
this.submitForm({cURL:url,cSubmitAs:submitAs});